# Blog Archives

## Day 159: AP Exam Day is Done!

I can’t wait to hear how my students did today! I am so proud of their focus and perseverance in preparing for this exam. There were 19 of them and only a few were a little nervous this morning. I made a support sign and hung it in the room the test was being given.

I saw many ideas for making a support package for standardized testing on Pinterest. I’ve wanted to do this for my AP Stats statletes for a few years now, but never seemed to find the time to actually put it together until this year. So last night, I gathered the requisite items and began to stuff the Care Packages (oops, there’s my take-out teriyaki in the mix).

The day before I looked at a variety of ideas and merged them into one for statistics. I thought my little anecdotes were quite qwippy…but I plan to make the sayings even better next year!

I am really glad that I took the time this year to put together my AP Statistics Exam Care Package. My students were very appreciative, too. I just love ’em to death! What do you think?

Unfortunately, we had a 90-minute block class just before the exam, and I didn’t want to have them staring at me while those not taking the exam “goofed around,” which would make the AP test-takers even more nervous than they already were feeling. So I decided to use the AP Statistics Murder Mystery activity, adapted by Linda Gann from Major Revak of the US Air Force Academy and offered at the AP Stats Best Practices evening. This was the first time I’ve used it and I think it was a nice, low-key way to do something “statsy” without focusing on the test. Most students found the activity relaxing too. But I’m still not sure it was the best thing just before the exam. Have to reflect on that some more! What ideas do you have for this kind of situation?

## Day 128: Revisiting Linear Motion

How do you connect “old” ideas (that is, previous learning) with “new” or unfamiliar situations? I’m always trying to build new dendrite connections between the old and the new. Linear motion via parametrics is just one example. As our opening activity, I asked students this question: *Write the vector equation and the parametric equations of the line through (-7, 4) and perpendicular to 2x + 3y = 12. Draw a diagram of the situation. *

I purposefully left out the t-interval to force a discussion of why it is important and how do we write parametric equations when there is no clear one-unit increment of time. And what a rich discussion we had! I found out what “stuck” from yesterday’s Geogebra exploration, drew in vectors (yay for review!) and really looked deeply at the variety of parametric equations that are correct even if they look different.

I was soooo thrilled when two students shared different x(t) equations (green vs. red). Immediately, students took sides (Creating a fight, as Dan Meyer would say).

As a class we compared and contrasted what was written: same starting x-coordinate, different rates of change. This led to a discussion of how one was 2 and the other was 7….it came down to the arbitrarily chosen “end” point. One student used the slope and found the next point along the line; the other student used the y-intercept. Cool, huh?! This drove us to explore whether both could be correct and why. Can we use both to determine a specific location, and how are they related. You can see the results of our queries.

How have you facilitated a deep discussion based on student responses? And what turned the discussion into a deep one?

## Day 109: I Love Lucy Visits Type 1/Type II Errors

We had fun today with I Love Lucy and Errors in AP Stats today. As an opening activity, we watched this snippet of I Love Lucy in the Chocolate Factory (thanks to a suggestion from one of our administrators who used to teach AP Stats):

Then I asked the following questions: What type of error did the supervisor make? What is the null and alternative hypotheses under which the supervisor was working? Be sure to define your parameter of interest.

Lucy is always funny and it lightened up the mood, got the kiddos talking about the errors and deciding what the null and alternative hypotheses might be in a real situation. Some students used the average speed of the belt and others used the proportion of candies that were wrapped to quantify what they saw and what the supervisor might have been using as her measure. Interestingly, with the first situation a Type I error was committed and in the other, a Type II error was committed.

Because of this quick video, a student asked if we can use either a mean or a proportion if we wanted to in a situation. This was a great opportunity to revisit the Organizing Data component of the AP curriculum, specifically that one item/subject can have many measurements.

I also got a chance to revisit my Inference Procedure Rubric. My students wrote up their second Significance Test today for the question:

According to an article in the

San Gabriel Valley Tribune(2-13-03), “Most people are kissing the ‘right way’.” That is, according to the study, the majority of couples tilt their heads to the right when kissing. In the study, a researcher observed a random sample 124 couples kissing in various public places and found that 83/124 (66.9%) of the couples tilted to the right. Is this convincing evidence that couples really do prefer to kiss the right way? Explain.

After they finished, I posted the rubric on the board and then toggled between the rubric and the sample answer. I had them assess their writing at each step. It also gave me the opportunity to reiterate typical errors I had already seen them make or errors I had seen previous students make. Great discussions! I think I will add the question: What type of error might the researcher have made?

Here are some student responses. In reflecting on their successes, it seems the typical misunderstanding about when to reject vs. not reject based on the p-value. I’m really happy that so few are still thinking a small p-value means the null hypothesis is supported. I also am really pleased with the number of students who fixed their errors as we went along (75% of them) and wrote notes to themselves about misconceptions.

In particular, its clear we need to continue to practice writing conclusion statements

As students were walking out the door, I asked how they were feeling about their confidence in writing up Significance Tests. Most said they felt much more confident overall and they were able to articulate clearly areas they need to practice (mostly writing up the conclusions.)

## Day 96: Error Analysis with Quick Poll

I used the Navigator today in precalculus to assess their ability to use logarithm properties and techniques. Once the problem was completed, it was submitted through Navigator. Then we looked at the results as a class, deciding if an answer was correct, even if the form of the answer was different. The green indicates correct and grey is not. I then save the results to the Portfolio and use it as a formative assessment score for participation.

This question below was interesting because we talked about the difference between “undefined” and “no solution” which leads back to the leveraging difference between an expression and equation. After the discussion, I agreed to accept both since “they didn’t know” but from now on I wouldn’t accept no solution.

Some kids are so creative in their final answer form, trying to outdo others with the most complicated yet correct answer. After a couple, I had to put a halt to their creativity….but it was fun. They asked for more, too.

## Day 66: Oh What We Can Do With Trig Functions!

Today in Precalculus, I had my students explore what happens when they encounter things like f(x) = |sin x|, g(x) = 2x + 3 cos 6x, and h(x) = (2x + 3)cos 6x.

Their job today was to work with various situations, make conjectures, test them and eventually make a generalization about what they are seeing. Eventually I would like them to connect the equation and what is actually happening graphically.

## Day 65: Random Variables and Gallery Walks

We are finishing up the introduction to Random Variables in AP Statistics today and I thought it was time for students to work together on some problems on discrete and continuous random variables including finding and interpreting the expected value and standard deviation of discrete RVs and using notation properly. Once again, I want the kids to be actively engaged in thinking about and doing the problems rather than passively watching me or another student work thorough a problem. Don’t get me wrong; there is a time and place for modeling good mathematical techniques, but we are past that today.

So I gave a 20 Minute Poster assignment. Groups were given 20 minutes to solve and write up a problem by applying their statistical knowledge and using good AP-level communication. Each member of the group chose one colored marker and could only write in their color on the poster. They signed their name in the same color so I could see the contributions made by each individual. Lots of good clarifying questions and discussion in making the poster: “Does a uniform distribution look like this (showing a normal curve)…then what does it look like?”, “why is P(X=3) equal zero in a continuous random variable distribution?”, “I don’t get why P(X>3) = P(X≥3) for a continuous random variable, but they aren’t equal for a discrete random variable,” and “how do we find the standard deviation if we don’t have the probability distribution given to us?”

After the posters were completed, I had students pair up using Clock Buddies…today it was their 7 o’clock buddies. Using a Gallery Walk protocol, the pairs had 5 minutes to read over and check the work on a poster that was not either of theirs, leaving sticky notes with “I notice…” and “I wonder…” comments regarding the correct application of statistical techniques and good communication. After 5 minutes, they moved to another problem and did the same. I will post the original problems and photos of their solutions on our website.

How do you use formative assessment techniques in your classes?

## Day 64: Ferris Wheel Matching

In precalculus today, it was time for students to work with a real-world application of sinusoidal functions while also engaging is sense-making discussion with their peers. I used the matching activity in an awesome Ferris Wheel Project I found at Mathematics Assessment Project: Assessing 21st Century Mathematics. The MAP project is a collaboration between the Shell Center team at the University of Nottingham and the University of California, Berkeley. There are lots of great Common Core based authentic lessons, so take a look.

The matching activity included three sets of cards: *Card Set A: Graphs*, *Card Set B: Functions* and *Card Set C: Descriptions of Wheels.* The students cut out the three sets of cards and began to match them. Interestingly, there were only 6 descriptions but 8 graphs and 7 equations (with a blank one for one graph that did not have a predetermined equation).

The discussions were fabulous and the support materials give teachers a guide for listening to student discussions and possible prompts and interventions that help but don’t inhibit the students’ sense-making. Many kids were up out of their seats to work together.

Many student groups thought they had the matching figured out and then realized that they had an equation that didn’t fit the graph after all and had to re-think their strategy.

The activity took approximately 30 minutes. I had an photo of the answers so the activity was also a “self-check.” Fun times and good learning took place.

## Day 63: Clock Buddies!

Last week I was talking with the Spanish teacher here in our building and she shared that she has kids pair up using clock buddies, in Spanish of course! What a great reminder that I had not set this up with my classes this year and I love the flexibility it offers, the ownership it gives to the students to pick their partners and guarantees variety of student pairings too.

So how does this student collaboration strategy work? Well, clock buddies is a variation on the “Speed Dating” strategy, but not so intensive. I hand out this clock buddies form I found at Reading Quest that has been reduced to half size. Using the clock image with a blank next to each time, students find a unique partner for each time slot. For instance, when student A finds a partner for, say, 2 o’clock, student A writes student B’s name on the 2 o’clock blank and student B writes student A’s name on their 2 o’clock blank. Every blank is to be filled with a different name. Occasionally, students can’t find a unique partner so we do a call-out and pair them up. If there is not a unique person for the time slot, then the student can use a name twice. Also, if there is an odd person out, then either they fill in when a clock buddy pair has an individual absent or I allow them to chose the clock buddy pair to join.

Once all of the cards are filled out, I have the students take a photo of it and store it on their iPad in their Notability app. If your students can’t take a photo, have them store their completed form in their notebook. What strategies do you use to get students talking to each other?

## Day 41: Students take over

One of the ways I am trying to ensure long-term retention as well as student discourse is through engaging opener problems and exit questions. Students submit final documents on Fridays through Schoology (love this feature of LMS). Today, I used a problem I found on Bob Lochel’s blog mathcoachblog. Since he used it with his Algebra 2 class, I figured it would be a good out-of-the-blue, retention/review question and it was! At first my students said things like, “I don’t know how to do this!,” and “We haven’t done a problem like this yet.” Of course I encouraged them, saying “of course you can do this…you know I don’t ask you questions that you can’t figure out with what you know.” Some grabbed their calculators, some expanded the factorials and others noticed the underlying structure. Great paired discussions. I then had students share their approach based on the complexity and eloquence of their solutions. I loved seeing the students listening to each other and offering suggestions and asking clarifying questions And I never did say what the “correct” answer was 🙂 However, I did point out that the underlying structure helped many students to get to the answer with ease and without a calculator…think before you begin a “brute force” approach.